Analysis of Trusses by Method of Joint

 

Analysis of Trusses by Method of Joint

This method is based on the principle that if a structural system constitutes a body in equilibrium, then any joint in that system is also in equilibrium and, thus, can be isolated from the entire system and analyzed using the conditions of equilibrium. The method of joint involves successively isolating each joint in a truss system and determining the axial forces in the members meeting at the joint by applying the equations of equilibrium. The detailed procedure for analysis by this method is stated below.

Procedure for Analysis

•Verify the stability and determinacy of the structure. If the truss is stable and determinate, then proceed to the next step.

•Determine the support reactions in the truss.

•Identify the zero-force members in the system. This will immeasurably reduce the computational efforts involved in the analysis.

•Select a joint to analyze. At no instance should there be more than two unknown member forces in the analyzed joint.

•Draw the isolated free-body diagram of the selected joint, and indicate the axial forces in all members meeting at the joint as tensile (i.e. as pulling away from the joint). If this initial assumption is wrong, the determined member axial force will be negative in the analysis, meaning that the member is in compression and not in tension.

•Apply the two equations ΣFX=0 and ΣFY=0 to determine the member axial forces.

•Continue the analysis by proceeding to the next joint with two or fewer unknown member forces.

Example 1

Using the method of joint, determine the axial force in each member of the truss shown in Figure

Fig.5.10. Truss.

 

Solution

Support reactions. By applying the equations of static equilibrium to the free-body diagram shown in Figure 5.10b, the support reactions can be determined as follows:

+↶∑MA=020(4)−12(3)+(8)Cy=0

Cy=−5.5kN+↑∑Fy=0

Ay−5.5+20=0

Ay=−14.5kN+→∑Fx=0−Ax+12=0

Ax=12kN

Cy=5.5kN↓

Ay=14.5kN↓

Ax=12kN←

Analysis of joints. The analysis begins with selecting a joint that has two or fewer unknown member forces. The free-body diagram of the truss will show that joints A and B satisfy this requirement. To determine the axial forces in members meeting at joint A, first isolate the joint from the truss and indicate the axial forces of members as FAB and FAD_, as shown in Figure 5.10c. The two unknown forces are initially assumed to be tensile (i.e. pulling away from the joint). If this initial assumption is incorrect, the computed values of the axial forces will be negative, signifying compression.

Analysis of joint A.

+↑∑Fy=0

FABsin36.87∘−14.5=0

FAB=24.17+→∑Fx=0−12+FAD+FABcos36.87∘=0

FAD=12−24.17cos36.87∘=−7.34kN

After completing the analysis of joint A, joint B or D can be analyzed, as there are only two unknown forces.

Analysis of joint D.

+↑∑Fy=0FDB=0+→∑Fx=0−FDA+FDC=0FDC=FDA=−7.34kN

Analysis of joint B.

+→∑Fx=0−FBAsin53.13+FBCsin53.13+15=0FBCsin53.13=−15+24.17sin53.13=FBC=5.42kN

5.6.3 Zero Force Members

Complex truss analysis can be greatly simplified by first identifying the “zero force members.” A zero force member is one that is not subjected to any axial load. Sometimes, such members are introduced into the truss system to prevent the buckling and vibration of other members. The truss-member arrangements that result in zero force members are listed as follows:

1.If noncollinearity exists between two members meeting at a joint that is not subjected to any external force, then the two members are zero force members (see Figure 5.11a).

2.If three members meet at a joint with no external force, and two of the members are collinear, the third member is a zero force member (see Figure 5.11b).

3.If two members meet at a joint, and an applied force at the joint is parallel to one member and perpendicular to the other, then the member perpendicular to the applied force is a zero force member (see Figure 5.11c).

Fig.5.11. Zero force members.